Find the largest integer N for which N−6 evenly divides N^3−6.

Answer 1

N^3−6 is zero Modulo (N-6). If we compute Modulo (N-6) then obviously:

N-6 = 0 ----->

N = 6

Here and in the following the equals sign means equality modulo N - 6.

We then have:

N^3 -6 = 6^3 - 6 = 210

Therefore:

210 = 0

Reverting back to the ordinary definition of the equals sign, this means that:

210 = k (N-6)

So, N-6 must be the largest possible factor of 210, which is 210 therefore
N = 216.